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An approximate universal coefficient theorem

2005
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Transactions of the American Mathematical Society
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An

doi:10.1090/s0002-9947-05-03696-2
fatcat:jr5272drsnemtm7uqmw4tyvn7m
*approximate**Universal*Coefficient*Theorem*(AUCT) for certain C * -algebras is established. ... We also show that C * -algebras that are locally*approximated*by C * -algebras satisfying the AUCT satisfy the AUCT. ... Part of this work was done when the author was visiting East China Normal*University*. He thanks the Department of Mathematics for providing a shelter during the summer 2000 in Shanghai. ...##
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Universal Approximation Theorem for Neural Networks
[article]

2021
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arXiv
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pre-print

Is there any theoretical guarantee for the

arXiv:2102.10993v1
fatcat:i3rbo7bs2jgvzlwhieapetitdi
*approximation*ability of neural networks? The answer to this question is the "*Universal**Approximation**Theorem*for Neural Networks". ... This paper is a comprehensive explanation of the*universal**approximation**theorem*for feedforward neural networks, its*approximation*rate problem (the relation between the number of intermediate units and ...##
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Universal approximation theorem for Dirichlet series

2006
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International Journal of Mathematics and Mathematical Sciences
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Several density results are proved that finally lead to the main

doi:10.1155/ijmms/2006/37014
fatcat:737d3r4ktzdxrb4547e3jz6qt4
*theorem*on simultaneous*approximation*. ... The paper deals with an extension*theorem*by Costakis and Vlachou on simultaneous*approximation*for holomorphic function to the setting of Dirichlet series, which are absolutely convergent in the right ... The present paper can also be seen as an extension of [6] to*universal*Dirichlet series. Moreover,*Theorem*4.3 is a*universal**approximation*result in the sense of [7] . ...##
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Mergelyan's approximation theorem with nonvanishing polynomials and universality of zeta-functions

2013
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Journal of Approximation Theory
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We prove a variant of the Mergelyan

doi:10.1016/j.jat.2012.12.005
fatcat:zelqzp3lfnakbfooevr6lvs5pq
*approximation**theorem*that allows us to*approximate*functions that are analytic and nonvanishing in the interior of a compact set K with connected complement, and whose ... We apply this result on the Voronin*universality**theorem*for compact sets K of this type, where the usual condition that the function is nonvanishing on the boundary can be removed. ... By Conjecture 2 we can*approximate*f (z) by a polynomial p(z) such that Relating Mergelyan's*theorem*and Voronin*universality*|p(z) − f (z)| < ε/2, (z ∈ K), (1) where p(z) is nonvanishing on K. ...##
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A Universal Approximation Theorem of Deep Neural Networks for Expressing Probability Distributions
[article]

2020
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arXiv
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pre-print

This paper studies the

arXiv:2004.08867v3
fatcat:bassh6lnu5czljl746ratj3eey
*universal**approximation*property of deep neural networks for representing probability distributions. ... We prove upper bounds for the size (width and depth) of the deep neural network in terms of the dimension d and the*approximation*error ε with respect to the three discrepancies. ... Our main result is the*universal**approximation**theorem*for expressing probability distributions.*Theorem*2.1 (Main*theorem*). ...##
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Universal Approximation Theorems for Differentiable Geometric Deep Learning
[article]

2022
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arXiv
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pre-print

In the Euclidean setting, our results imply a quantitative version of Kidger and Lyons (2020)'s

arXiv:2101.05390v4
fatcat:dgbgiv7ymrbttapiwlgiizrtaa
*approximation**theorem*and a data-dependent version of Yarotsky and Zhevnerchuk (2019)'s uncursed*approximation*... As applications, we confirm the*universal**approximation*capabilities of the following GDL models: Ganea et al. (2018)'s hyperbolic feedforward networks, the architecture implementing Krishnan et al. (2015 ... These networks are obtained via the*universal**approximation**theorem*. Hence, to derive the estimates of Proposition 53, we should constructively prove the*universal**approximation**theorem*(*Theorem*46). ...##
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The universal approximation theorem for complex-valued neural networks
[article]

2020
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arXiv
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pre-print

We generalize the classical

arXiv:2012.03351v1
fatcat:3wq474t47vctre3oidrbpugo6a
*universal**approximation**theorem*for neural networks to the case of complex-valued neural networks. ... We completely characterize those activation functions σ for which the associated complex networks have the*universal**approximation*property, meaning that they can uniformly*approximate*any continuous function ... Related work The classical*universal**approximation**theorem*There exist many versions of the*universal**approximation**theorem*for real networks. ...##
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Arbitrary-Depth Universal Approximation Theorems for Operator Neural Networks
[article]

2021
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arXiv
*
pre-print

The standard

arXiv:2109.11354v1
fatcat:rtezxzkrubglzikwjikbgqoeji
*Universal**Approximation**Theorem*for operator neural networks (NNs) holds for arbitrary width and bounded depth. ... Here, we prove that operator NNs of bounded width and arbitrary depth are*universal**approximators*for continuous nonlinear operators. ... In the*approximation*theory of neural networks (NNs),*universal**approximation**theorems*(UATs) are statements that establish the density of a class of NNs within a space of mappings. ...##
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A Universal Approximation Theorem for Mixture-of-Experts Models

2016
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Neural Computation
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Our result can be viewed as a

doi:10.1162/neco_a_00892
pmid:27626962
fatcat:7gwgc5ahezgenmcftmjyl4dxca
*universal**approximation**theorem*for MoE models. ... The*theorem*we present allows MoE users to be confident in applying such models for estimation when data arise from nonlinear and nondifferentiable generative processes. ... Our result is a*universal**approximation**theorem*, similar in spirit to Cybenko (1989,*theorem*2) , where the linear combination of sigmoidal functions is proved dense in C(X). ...##
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Universal approximation theorem for uninorm-based fuzzy systems modeling

2003
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Fuzzy sets and systems (Print)
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A Universal Approximation Theorem for Mixture of Experts Models
[article]

2016
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arXiv
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pre-print

Our result can be viewed as a

arXiv:1602.03683v1
fatcat:svv6rphry5fdjfj2rfqlujjjo4
*universal**approximation**theorem*for MoE models. ... Our result is a*universal**approximation**theorem*for MoE mean functions in the style of [3] . ... Stone-Weierstrass*Theorem*Following the presentation of [2] , the Stone-Weierstrass*Theorem*can be phrased as follows.*Theorem*1. ...##
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Lavrentiev's approximation theorem with nonvanishing polynomials and universality of zeta-functions
[article]

2010
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arXiv
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pre-print

We prove a variant of the Lavrentiev's

arXiv:1010.0386v1
fatcat:e3rwnxjmpnekphxsjajhflbdmi
*approximation**theorem*that allows us to*approximate*a continuous function on a compact set K in C without interior points and with connected complement, with polynomial ... We use this result to obtain a version of the Voronin*universality**theorem*for compact sets K, without interior points and with connected complement where it is sufficient that the function is continuous ... By*Theorem*1 we can*approximate*f (z) by a polynomial g(z) such that |g(z) − f (z)| < ǫ/2, z ∈ K, (3) where g(z) is nonvanishing on K. ...##
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Characterizing the Universal Approximation Property
[article]

2020
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arXiv
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pre-print

Moreover, we show that most function spaces admit

arXiv:1910.03344v3
fatcat:j3mhfpj3mbf35iz2ypotvyt5de
*universal**approximators*built using a single function. ... To better understand the*approximation*capabilities of various currently available neural network architectures, this paper studies the*universal**approximation*property itself across a broad scope of function ...*Theorem*2.3 (Equivalence of*Universal**Approximators*to the Feed-Forward Architecture). ...##
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DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
[article]

2020
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arXiv
*
pre-print

This

arXiv:1910.03193v3
fatcat:67kretzwczffriihix3zn7fs6i
*universal**approximation**theorem*is suggestive of the potential application of neural networks in learning nonlinear operators from data. ... While it is widely known that neural networks are*universal**approximators*of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can ... Acknowledgments We thank Yanhui Su of Fuzhou*University*for the help on*Theorem*2. We thank Zhongqiang Zhang of Worcester Polytechnic Institute for the proof in Appendix C. ...##
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A Gradient Free Neural Network Framework Based on Universal Approximation Theorem
[article]

2020
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arXiv
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pre-print

We present a numerical scheme for computation of Artificial Neural Networks (ANN) weights, which stems from the

arXiv:1909.13563v3
fatcat:5u34jsdskvho7e5bc2fj3u6rai
*Universal**Approximation**Theorem*, avoiding laborious iterations. ... The method is based on the calculation of the weights of each neuron in small neighborhoods of the data, such that the corresponding local*approximation*matrix is invertible. ... This complies with the*Universal**Approximation**theorem*, and offers a geometric point of view. ...
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